Was it a mistake adopting kitten with older cat

[CatsQuestion]

I don’t know if I’ve made a mistake, or if I need to give it some time. We’ve had our new kitten just over a week, she was a rehome as the owner could not cope, she’s 6 months old, not spayed, as the previous owner didn’t bother but is due to be done mid-December, that’s the only time the vet could fit us in. Our other cat is 12 in January. They both seem to tolerate each other but every night and morning, our kitten constantly chases our older cat and she’s spending more time hiding under our bed and going outside whereas before she was mostly indoor, if we keep the kitten in the kitchen she will mew constantly and scratch the door throughout the night and she has mastered how to open the door to get out. I always play with her for 20 minutes each night, she’s around me during the day too. This morning they were both up on their hind legs, pawing at each other, I’m not sure if they were playing or if fighting, as there were a few loud meows whilst this happened.

Any advice please?

Imagine being retired but then suddenly having a toddler move in.
You want to sit and rest but all you get is “Play with me! Play with me! Play with me! endlessly. You’d be grumpy or find reasons to be out of the house too I imagine.
You say you play with the kitten 20 mins at night, one play session really isn’t enough. She may be around you during the day but maybe a few more play sessions would mean your older cat isn’t bothered to the point of annoyance and they will settle together a bit happier?

The amount of playtime your kitten needs will vary from kitten to kitten, as some cats are more energetic than others, but they generally need at least two to three play sessions a day, if not more

Play with your kitten at least two to three times a day (ideally for 15 minutes at a time) using interactive toys. These are toys that you move and they chase.

We did this last year but we had to keep them apart for a few weeks before introducing them, so maybe do that. It did work but it does take time and mine ended up being great buddies (once the older cat made sure kitten knew who was still the boss ;)

icatcare.org/advice/introducing-an-adult-cat-to-your-cat/ - this was the guidelines that we followed

Going cats are demanding she needs lots of love and playtime, my old puss 12, was a handful at 18 months
next time older cat

It does take time, and it's a balance between bonding, and allowing older cat peace and quiet.
I have four cats, one is 8, he lost his brother earlier in in year. A few months ago l rehomed 2 kittens, but sadly lost one, she died at the vets during spay repair operation. Her sibling seemed lost, my older cat wasn't interested in him, so l got two more kittens, the 3 younger ones now get on like a house on fire, the older one sleeps and eats in same room as them but rest of day does his own thing, he seems better with the older kitten, but slowly coming around. I don't force anything, house and garden is big enough for four cats, eldest one stayed out more to begin with but now coming in a lot more. I make sure he has plenty of fuss, treats and his own space. The youngest ones entertain themselves, l do play with them, and they go outdoors but they don't bother my older cat, l think having one kitten on its own is hard work, my 3 younger ones play chase, groom one another and sleep snuggled up, they are inseparable. All four are currently fast asleep in same room.